Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Percentage Accurate: 89.0% → 96.9%

Time: 20.6s

Alternatives: 19

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ t_2 := \frac{y}{x + 1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{t\_2 - \frac{x}{z \cdot \left(x + 1\right)}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)))
        (t_2 (/ y (+ x 1.0))))
   (if (<= t_1 (- INFINITY))
     (* (/ z (fma t z (- x))) t_2)
     (if (<= t_1 1e+307)
       t_1
       (+ (/ x (+ x 1.0)) (/ (- t_2 (/ x (* z (+ x 1.0)))) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double t_2 = y / (x + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (z / fma(t, z, -x)) * t_2;
	} else if (t_1 <= 1e+307) {
		tmp = t_1;
	} else {
		tmp = (x / (x + 1.0)) + ((t_2 - (x / (z * (x + 1.0)))) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_2 = Float64(y / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(z / fma(t, z, Float64(-x))) * t_2);
	elseif (t_1 <= 1e+307)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(t_2 - Float64(x / Float64(z * Float64(x + 1.0)))) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(z / N[(t * z + (-x)), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 1e+307], t$95$1, N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[(x / N[(z * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -inf.0

   1. Initial program 38.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 38.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 38.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 37.2%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   6. Step-by-step derivation

      1. times-frac 88.3%

         \[\leadsto \color{blue}{\frac{y}{1 + x} \cdot \frac{z}{t \cdot z - x}} \]

      2. +-commutative 88.3%

         \[\leadsto \frac{y}{\color{blue}{x + 1}} \cdot \frac{z}{t \cdot z - x} \]

      3. fma-neg 88.3%

         \[\leadsto \frac{y}{x + 1} \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(t, z, -x\right)}} \]

   7. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{y}{x + 1} \cdot \frac{z}{\mathsf{fma}\left(t, z, -x\right)}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 9.99999999999999986e306

   1. Initial program 98.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   if 9.99999999999999986e306 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 6.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 6.7%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 6.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 87.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t} + \frac{x}{1 + x}} \]
   6. Step-by-step derivation

      1. +-commutative 87.6%

         \[\leadsto \color{blue}{\frac{x}{1 + x} + -1 \cdot \frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t}} \]

      2. mul-1-neg 87.6%

         \[\leadsto \frac{x}{1 + x} + \color{blue}{\left(-\frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t}\right)} \]

      3. unsub-neg 87.6%

         \[\leadsto \color{blue}{\frac{x}{1 + x} - \frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t}} \]

      4. +-commutative 87.6%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} - \frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t} \]

      5. sub-neg 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\color{blue}{-1 \cdot \frac{y}{1 + x} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}}{t} \]

      6. mul-1-neg 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\color{blue}{\left(-\frac{y}{1 + x}\right)} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      7. distribute-neg-frac2 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\color{blue}{\frac{y}{-\left(1 + x\right)}} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      8. distribute-neg-in 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{\color{blue}{\left(-1\right) + \left(-x\right)}} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      9. metadata-eval 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{\color{blue}{-1} + \left(-x\right)} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      10. unsub-neg 87.6%

          \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{\color{blue}{-1 - x}} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      11. mul-1-neg 87.6%

          \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{-1 - x} + \left(-\color{blue}{\left(-\frac{x}{z \cdot \left(1 + x\right)}\right)}\right)}{t} \]

      12. remove-double-neg 87.6%

          \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{-1 - x} + \color{blue}{\frac{x}{z \cdot \left(1 + x\right)}}}{t} \]

      13. +-commutative 87.6%

          \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{-1 - x} + \frac{x}{z \cdot \color{blue}{\left(x + 1\right)}}}{t} \]

   7. Simplified 87.6%

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{\frac{y}{-1 - x} + \frac{x}{z \cdot \left(x + 1\right)}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(t, z, -x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+307}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{\frac{y}{x + 1} - \frac{x}{z \cdot \left(x + 1\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{t\_1 \cdot \left(x + 1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{\frac{y}{x + 1} - \frac{x}{z \cdot \left(x + 1\right)}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 (- INFINITY))
     (* y (/ z (* t_1 (+ x 1.0))))
     (if (<= t_2 1e+307)
       t_2
       (+ (/ x (+ x 1.0)) (/ (- (/ y (+ x 1.0)) (/ x (* z (+ x 1.0)))) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y * (z / (t_1 * (x + 1.0)));
	} else if (t_2 <= 1e+307) {
		tmp = t_2;
	} else {
		tmp = (x / (x + 1.0)) + (((y / (x + 1.0)) - (x / (z * (x + 1.0)))) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (z / (t_1 * (x + 1.0)));
	} else if (t_2 <= 1e+307) {
		tmp = t_2;
	} else {
		tmp = (x / (x + 1.0)) + (((y / (x + 1.0)) - (x / (z * (x + 1.0)))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = y * (z / (t_1 * (x + 1.0)))
	elif t_2 <= 1e+307:
		tmp = t_2
	else:
		tmp = (x / (x + 1.0)) + (((y / (x + 1.0)) - (x / (z * (x + 1.0)))) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / Float64(t_1 * Float64(x + 1.0))));
	elseif (t_2 <= 1e+307)
		tmp = t_2;
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(Float64(y / Float64(x + 1.0)) - Float64(x / Float64(z * Float64(x + 1.0)))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = y * (z / (t_1 * (x + 1.0)));
	elseif (t_2 <= 1e+307)
		tmp = t_2;
	else
		tmp = (x / (x + 1.0)) + (((y / (x + 1.0)) - (x / (z * (x + 1.0)))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y * N[(z / N[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], t$95$2, N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(z * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -inf.0

   1. Initial program 38.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 38.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 38.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 37.2%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 71.1%

         \[\leadsto \color{blue}{y \cdot \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. +-commutative 71.1%

         \[\leadsto y \cdot \frac{z}{\color{blue}{\left(x + 1\right)} \cdot \left(t \cdot z - x\right)} \]

      3. fma-neg 71.1%

         \[\leadsto y \cdot \frac{z}{\left(x + 1\right) \cdot \color{blue}{\mathsf{fma}\left(t, z, -x\right)}} \]

   7. Simplified 71.1%

      \[\leadsto \color{blue}{y \cdot \frac{z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}} \]

   8. Taylor expanded in y around 0 37.2%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* 71.1%

         \[\leadsto \color{blue}{y \cdot \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. +-commutative 71.1%

         \[\leadsto y \cdot \frac{z}{\color{blue}{\left(x + 1\right)} \cdot \left(t \cdot z - x\right)} \]

   10. Simplified 71.1%

       \[\leadsto \color{blue}{y \cdot \frac{z}{\left(x + 1\right) \cdot \left(t \cdot z - x\right)}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 9.99999999999999986e306

   1. Initial program 98.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   if 9.99999999999999986e306 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 6.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 6.7%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 6.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 87.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t} + \frac{x}{1 + x}} \]
   6. Step-by-step derivation

      1. +-commutative 87.6%

         \[\leadsto \color{blue}{\frac{x}{1 + x} + -1 \cdot \frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t}} \]

      2. mul-1-neg 87.6%

         \[\leadsto \frac{x}{1 + x} + \color{blue}{\left(-\frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t}\right)} \]

      3. unsub-neg 87.6%

         \[\leadsto \color{blue}{\frac{x}{1 + x} - \frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t}} \]

      4. +-commutative 87.6%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} - \frac{-1 \cdot \frac{y}{1 + x} - -1 \cdot \frac{x}{z \cdot \left(1 + x\right)}}{t} \]

      5. sub-neg 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\color{blue}{-1 \cdot \frac{y}{1 + x} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}}{t} \]

      6. mul-1-neg 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\color{blue}{\left(-\frac{y}{1 + x}\right)} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      7. distribute-neg-frac2 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\color{blue}{\frac{y}{-\left(1 + x\right)}} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      8. distribute-neg-in 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{\color{blue}{\left(-1\right) + \left(-x\right)}} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      9. metadata-eval 87.6%

         \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{\color{blue}{-1} + \left(-x\right)} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      10. unsub-neg 87.6%

          \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{\color{blue}{-1 - x}} + \left(--1 \cdot \frac{x}{z \cdot \left(1 + x\right)}\right)}{t} \]

      11. mul-1-neg 87.6%

          \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{-1 - x} + \left(-\color{blue}{\left(-\frac{x}{z \cdot \left(1 + x\right)}\right)}\right)}{t} \]

      12. remove-double-neg 87.6%

          \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{-1 - x} + \color{blue}{\frac{x}{z \cdot \left(1 + x\right)}}}{t} \]

      13. +-commutative 87.6%

          \[\leadsto \frac{x}{x + 1} - \frac{\frac{y}{-1 - x} + \frac{x}{z \cdot \color{blue}{\left(x + 1\right)}}}{t} \]

   7. Simplified 87.6%

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{\frac{y}{-1 - x} + \frac{x}{z \cdot \left(x + 1\right)}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+307}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{\frac{y}{x + 1} - \frac{x}{z \cdot \left(x + 1\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t\_1}}{x + 1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{t\_1 \cdot \left(x + 1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 (- INFINITY))
     (* y (/ z (* t_1 (+ x 1.0))))
     (if (<= t_2 1e+307) t_2 (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y * (z / (t_1 * (x + 1.0)));
	} else if (t_2 <= 1e+307) {
		tmp = t_2;
	} else {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (z / (t_1 * (x + 1.0)));
	} else if (t_2 <= 1e+307) {
		tmp = t_2;
	} else {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = y * (z / (t_1 * (x + 1.0)))
	elif t_2 <= 1e+307:
		tmp = t_2
	else:
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / Float64(t_1 * Float64(x + 1.0))));
	elseif (t_2 <= 1e+307)
		tmp = t_2;
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = y * (z / (t_1 * (x + 1.0)));
	elseif (t_2 <= 1e+307)
		tmp = t_2;
	else
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y * N[(z / N[(t$95$1 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], t$95$2, N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -inf.0

   1. Initial program 38.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 38.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 38.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 37.2%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 71.1%

         \[\leadsto \color{blue}{y \cdot \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. +-commutative 71.1%

         \[\leadsto y \cdot \frac{z}{\color{blue}{\left(x + 1\right)} \cdot \left(t \cdot z - x\right)} \]

      3. fma-neg 71.1%

         \[\leadsto y \cdot \frac{z}{\left(x + 1\right) \cdot \color{blue}{\mathsf{fma}\left(t, z, -x\right)}} \]

   7. Simplified 71.1%

      \[\leadsto \color{blue}{y \cdot \frac{z}{\left(x + 1\right) \cdot \mathsf{fma}\left(t, z, -x\right)}} \]

   8. Taylor expanded in y around 0 37.2%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* 71.1%

         \[\leadsto \color{blue}{y \cdot \frac{z}{\left(1 + x\right) \cdot \left(t \cdot z - x\right)}} \]

      2. +-commutative 71.1%

         \[\leadsto y \cdot \frac{z}{\color{blue}{\left(x + 1\right)} \cdot \left(t \cdot z - x\right)} \]

   10. Simplified 71.1%

       \[\leadsto \color{blue}{y \cdot \frac{z}{\left(x + 1\right) \cdot \left(t \cdot z - x\right)}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 9.99999999999999986e306

   1. Initial program 98.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   if 9.99999999999999986e306 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 6.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 6.7%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 6.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 6.7%

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]

   6. Taylor expanded in t around inf 87.5%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 87.5%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} + \frac{y}{t \cdot \left(1 + x\right)} \]

      2. +-commutative 87.5%

         \[\leadsto \frac{x}{x + 1} + \frac{y}{t \cdot \color{blue}{\left(x + 1\right)}} \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{\left(z \cdot t - x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+307}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1}\\ t_2 := 1 - \frac{y \cdot z}{x}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-83}:\\ \;\;\;\;1 - z \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-160}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-172}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ x 1.0))) (t_2 (- 1.0 (/ (* y z) x))))
   (if (<= x -4.5e-14)
     t_1
     (if (<= x -2.65e-83)
       (- 1.0 (* z (/ y x)))
       (if (<= x -8.2e-131)
         t_1
         (if (<= x -8.2e-160)
           t_2
           (if (<= x -7.6e-172)
             t_1
             (if (<= x 9.8e-110) (/ y t) (if (<= x 6.2e-23) t_2 t_1)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double t_2 = 1.0 - ((y * z) / x);
	double tmp;
	if (x <= -4.5e-14) {
		tmp = t_1;
	} else if (x <= -2.65e-83) {
		tmp = 1.0 - (z * (y / x));
	} else if (x <= -8.2e-131) {
		tmp = t_1;
	} else if (x <= -8.2e-160) {
		tmp = t_2;
	} else if (x <= -7.6e-172) {
		tmp = t_1;
	} else if (x <= 9.8e-110) {
		tmp = y / t;
	} else if (x <= 6.2e-23) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / (x + 1.0d0)
    t_2 = 1.0d0 - ((y * z) / x)
    if (x <= (-4.5d-14)) then
        tmp = t_1
    else if (x <= (-2.65d-83)) then
        tmp = 1.0d0 - (z * (y / x))
    else if (x <= (-8.2d-131)) then
        tmp = t_1
    else if (x <= (-8.2d-160)) then
        tmp = t_2
    else if (x <= (-7.6d-172)) then
        tmp = t_1
    else if (x <= 9.8d-110) then
        tmp = y / t
    else if (x <= 6.2d-23) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double t_2 = 1.0 - ((y * z) / x);
	double tmp;
	if (x <= -4.5e-14) {
		tmp = t_1;
	} else if (x <= -2.65e-83) {
		tmp = 1.0 - (z * (y / x));
	} else if (x <= -8.2e-131) {
		tmp = t_1;
	} else if (x <= -8.2e-160) {
		tmp = t_2;
	} else if (x <= -7.6e-172) {
		tmp = t_1;
	} else if (x <= 9.8e-110) {
		tmp = y / t;
	} else if (x <= 6.2e-23) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (x + 1.0)
	t_2 = 1.0 - ((y * z) / x)
	tmp = 0
	if x <= -4.5e-14:
		tmp = t_1
	elif x <= -2.65e-83:
		tmp = 1.0 - (z * (y / x))
	elif x <= -8.2e-131:
		tmp = t_1
	elif x <= -8.2e-160:
		tmp = t_2
	elif x <= -7.6e-172:
		tmp = t_1
	elif x <= 9.8e-110:
		tmp = y / t
	elif x <= 6.2e-23:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x + 1.0))
	t_2 = Float64(1.0 - Float64(Float64(y * z) / x))
	tmp = 0.0
	if (x <= -4.5e-14)
		tmp = t_1;
	elseif (x <= -2.65e-83)
		tmp = Float64(1.0 - Float64(z * Float64(y / x)));
	elseif (x <= -8.2e-131)
		tmp = t_1;
	elseif (x <= -8.2e-160)
		tmp = t_2;
	elseif (x <= -7.6e-172)
		tmp = t_1;
	elseif (x <= 9.8e-110)
		tmp = Float64(y / t);
	elseif (x <= 6.2e-23)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (x + 1.0);
	t_2 = 1.0 - ((y * z) / x);
	tmp = 0.0;
	if (x <= -4.5e-14)
		tmp = t_1;
	elseif (x <= -2.65e-83)
		tmp = 1.0 - (z * (y / x));
	elseif (x <= -8.2e-131)
		tmp = t_1;
	elseif (x <= -8.2e-160)
		tmp = t_2;
	elseif (x <= -7.6e-172)
		tmp = t_1;
	elseif (x <= 9.8e-110)
		tmp = y / t;
	elseif (x <= 6.2e-23)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e-14], t$95$1, If[LessEqual[x, -2.65e-83], N[(1.0 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.2e-131], t$95$1, If[LessEqual[x, -8.2e-160], t$95$2, If[LessEqual[x, -7.6e-172], t$95$1, If[LessEqual[x, 9.8e-110], N[(y / t), $MachinePrecision], If[LessEqual[x, 6.2e-23], t$95$2, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.4999999999999998e-14 or -2.65e-83 < x < -8.2000000000000004e-131 or -8.20000000000000003e-160 < x < -7.59999999999999974e-172 or 6.1999999999999998e-23 < x

   1. Initial program 83.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 83.9%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 81.5%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   6. Step-by-step derivation

      1. +-commutative 81.5%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   7. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if -4.4999999999999998e-14 < x < -2.65e-83

   1. Initial program 99.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 63.5%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 64.1%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 64.1%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 64.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 64.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 64.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 64.1%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 64.1%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 64.1%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 64.1%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 64.1%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 64.1%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 64.1%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 64.1%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 64.1%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 64.1%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   9. Taylor expanded in x around 0 64.1%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}} \]
   10. Step-by-step derivation

       1. mul-1-neg 64.1%

          \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x}\right)} \]

       2. *-commutative 64.1%

          \[\leadsto 1 + \left(-\frac{\color{blue}{z \cdot y}}{x}\right) \]

       3. associate-*r/ 64.1%

          \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x}}\right) \]

       4. distribute-rgt-neg-in 64.1%

          \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x}\right)} \]

       5. distribute-neg-frac2 64.1%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{y}{-x}} \]

   11. Simplified 64.1%

       \[\leadsto 1 + \color{blue}{z \cdot \frac{y}{-x}} \]

   if -8.2000000000000004e-131 < x < -8.20000000000000003e-160 or 9.7999999999999995e-110 < x < 6.1999999999999998e-23

   1. Initial program 96.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 47.0%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 62.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 62.5%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 62.5%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 62.5%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 62.5%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 47.2%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 47.2%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 47.2%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 47.2%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 47.2%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 47.2%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 47.2%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 47.2%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 47.2%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 47.2%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   9. Taylor expanded in x around 0 62.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 62.5%

          \[\leadsto 1 + \color{blue}{\frac{-1 \cdot \left(y \cdot z\right)}{x}} \]

       2. neg-mul-1 62.5%

          \[\leadsto 1 + \frac{\color{blue}{-y \cdot z}}{x} \]

       3. distribute-rgt-neg-in 62.5%

          \[\leadsto 1 + \frac{\color{blue}{y \cdot \left(-z\right)}}{x} \]

   11. Simplified 62.5%

       \[\leadsto 1 + \color{blue}{\frac{y \cdot \left(-z\right)}{x}} \]

   if -7.59999999999999974e-172 < x < 9.7999999999999995e-110

   1. Initial program 87.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 87.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 81.2%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 81.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 81.2%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 81.2%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 81.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 81.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 81.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   10. Taylor expanded in x around 0 64.8%

       \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-83}:\\ \;\;\;\;1 - z \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-160}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-172}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ t_2 := \frac{x}{x + 1}\\ t_3 := 1 - \frac{y \cdot z}{x}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-82}:\\ \;\;\;\;1 - z \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-159}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-33}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ 1.0 (/ -1.0 (* z t)))))
        (t_2 (/ x (+ x 1.0)))
        (t_3 (- 1.0 (/ (* y z) x))))
   (if (<= x -1.7e-14)
     t_2
     (if (<= x -7.8e-82)
       (- 1.0 (* z (/ y x)))
       (if (<= x -1.5e-131)
         t_1
         (if (<= x -1.7e-159)
           t_3
           (if (<= x -4.9e-242)
             t_1
             (if (<= x 9e-110) (/ y t) (if (<= x 1.35e-33) t_3 t_2)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (1.0 + (-1.0 / (z * t)));
	double t_2 = x / (x + 1.0);
	double t_3 = 1.0 - ((y * z) / x);
	double tmp;
	if (x <= -1.7e-14) {
		tmp = t_2;
	} else if (x <= -7.8e-82) {
		tmp = 1.0 - (z * (y / x));
	} else if (x <= -1.5e-131) {
		tmp = t_1;
	} else if (x <= -1.7e-159) {
		tmp = t_3;
	} else if (x <= -4.9e-242) {
		tmp = t_1;
	} else if (x <= 9e-110) {
		tmp = y / t;
	} else if (x <= 1.35e-33) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = x * (1.0d0 + ((-1.0d0) / (z * t)))
    t_2 = x / (x + 1.0d0)
    t_3 = 1.0d0 - ((y * z) / x)
    if (x <= (-1.7d-14)) then
        tmp = t_2
    else if (x <= (-7.8d-82)) then
        tmp = 1.0d0 - (z * (y / x))
    else if (x <= (-1.5d-131)) then
        tmp = t_1
    else if (x <= (-1.7d-159)) then
        tmp = t_3
    else if (x <= (-4.9d-242)) then
        tmp = t_1
    else if (x <= 9d-110) then
        tmp = y / t
    else if (x <= 1.35d-33) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (1.0 + (-1.0 / (z * t)));
	double t_2 = x / (x + 1.0);
	double t_3 = 1.0 - ((y * z) / x);
	double tmp;
	if (x <= -1.7e-14) {
		tmp = t_2;
	} else if (x <= -7.8e-82) {
		tmp = 1.0 - (z * (y / x));
	} else if (x <= -1.5e-131) {
		tmp = t_1;
	} else if (x <= -1.7e-159) {
		tmp = t_3;
	} else if (x <= -4.9e-242) {
		tmp = t_1;
	} else if (x <= 9e-110) {
		tmp = y / t;
	} else if (x <= 1.35e-33) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (1.0 + (-1.0 / (z * t)))
	t_2 = x / (x + 1.0)
	t_3 = 1.0 - ((y * z) / x)
	tmp = 0
	if x <= -1.7e-14:
		tmp = t_2
	elif x <= -7.8e-82:
		tmp = 1.0 - (z * (y / x))
	elif x <= -1.5e-131:
		tmp = t_1
	elif x <= -1.7e-159:
		tmp = t_3
	elif x <= -4.9e-242:
		tmp = t_1
	elif x <= 9e-110:
		tmp = y / t
	elif x <= 1.35e-33:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(1.0 + Float64(-1.0 / Float64(z * t))))
	t_2 = Float64(x / Float64(x + 1.0))
	t_3 = Float64(1.0 - Float64(Float64(y * z) / x))
	tmp = 0.0
	if (x <= -1.7e-14)
		tmp = t_2;
	elseif (x <= -7.8e-82)
		tmp = Float64(1.0 - Float64(z * Float64(y / x)));
	elseif (x <= -1.5e-131)
		tmp = t_1;
	elseif (x <= -1.7e-159)
		tmp = t_3;
	elseif (x <= -4.9e-242)
		tmp = t_1;
	elseif (x <= 9e-110)
		tmp = Float64(y / t);
	elseif (x <= 1.35e-33)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (1.0 + (-1.0 / (z * t)));
	t_2 = x / (x + 1.0);
	t_3 = 1.0 - ((y * z) / x);
	tmp = 0.0;
	if (x <= -1.7e-14)
		tmp = t_2;
	elseif (x <= -7.8e-82)
		tmp = 1.0 - (z * (y / x));
	elseif (x <= -1.5e-131)
		tmp = t_1;
	elseif (x <= -1.7e-159)
		tmp = t_3;
	elseif (x <= -4.9e-242)
		tmp = t_1;
	elseif (x <= 9e-110)
		tmp = y / t;
	elseif (x <= 1.35e-33)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(1.0 + N[(-1.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e-14], t$95$2, If[LessEqual[x, -7.8e-82], N[(1.0 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-131], t$95$1, If[LessEqual[x, -1.7e-159], t$95$3, If[LessEqual[x, -4.9e-242], t$95$1, If[LessEqual[x, 9e-110], N[(y / t), $MachinePrecision], If[LessEqual[x, 1.35e-33], t$95$3, t$95$2]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.70000000000000001e-14 or 1.35e-33 < x

   1. Initial program 83.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 83.7%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 83.1%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   6. Step-by-step derivation

      1. +-commutative 83.1%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   7. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if -1.70000000000000001e-14 < x < -7.79999999999999947e-82

   1. Initial program 99.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 63.5%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 64.1%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 64.1%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 64.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 64.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 64.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 64.1%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 64.1%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 64.1%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 64.1%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 64.1%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 64.1%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 64.1%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 64.1%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 64.1%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 64.1%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   9. Taylor expanded in x around 0 64.1%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}} \]
   10. Step-by-step derivation

       1. mul-1-neg 64.1%

          \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x}\right)} \]

       2. *-commutative 64.1%

          \[\leadsto 1 + \left(-\frac{\color{blue}{z \cdot y}}{x}\right) \]

       3. associate-*r/ 64.1%

          \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x}}\right) \]

       4. distribute-rgt-neg-in 64.1%

          \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x}\right)} \]

       5. distribute-neg-frac2 64.1%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{y}{-x}} \]

   11. Simplified 64.1%

       \[\leadsto 1 + \color{blue}{z \cdot \frac{y}{-x}} \]

   if -7.79999999999999947e-82 < x < -1.49999999999999998e-131 or -1.69999999999999992e-159 < x < -4.90000000000000002e-242

   1. Initial program 89.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 89.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 64.1%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]

   6. Taylor expanded in x around 0 57.3%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{t \cdot z}\right)} \]

   if -1.49999999999999998e-131 < x < -1.69999999999999992e-159 or 9.0000000000000002e-110 < x < 1.35e-33

   1. Initial program 96.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 47.0%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 62.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 62.5%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 62.5%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 62.5%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 62.5%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 47.2%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 47.2%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 47.2%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 47.2%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 47.2%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 47.2%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 47.2%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 47.2%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 47.2%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 47.2%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   9. Taylor expanded in x around 0 62.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 62.5%

          \[\leadsto 1 + \color{blue}{\frac{-1 \cdot \left(y \cdot z\right)}{x}} \]

       2. neg-mul-1 62.5%

          \[\leadsto 1 + \frac{\color{blue}{-y \cdot z}}{x} \]

       3. distribute-rgt-neg-in 62.5%

          \[\leadsto 1 + \frac{\color{blue}{y \cdot \left(-z\right)}}{x} \]

   11. Simplified 62.5%

       \[\leadsto 1 + \color{blue}{\frac{y \cdot \left(-z\right)}{x}} \]

   if -4.90000000000000002e-242 < x < 9.0000000000000002e-110

   1. Initial program 85.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 85.9%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 85.9%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 82.1%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 82.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 82.0%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 82.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 82.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   10. Taylor expanded in x around 0 70.1%

       \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-82}:\\ \;\;\;\;1 - z \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-159}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-242}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-33}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{y \cdot z}{x}\\ t_2 := x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ t_3 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-14}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-82}:\\ \;\;\;\;1 + z \cdot \left(y - \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-131}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-242}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (* y z) x)))
        (t_2 (* x (+ 1.0 (/ -1.0 (* z t)))))
        (t_3 (/ x (+ x 1.0))))
   (if (<= x -4.8e-14)
     t_3
     (if (<= x -1.3e-82)
       (+ 1.0 (* z (- y (/ y x))))
       (if (<= x -1.35e-131)
         t_2
         (if (<= x -4.6e-159)
           t_1
           (if (<= x -4.9e-242)
             t_2
             (if (<= x 1.25e-109) (/ y t) (if (<= x 7.8e-39) t_1 t_3)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - ((y * z) / x);
	double t_2 = x * (1.0 + (-1.0 / (z * t)));
	double t_3 = x / (x + 1.0);
	double tmp;
	if (x <= -4.8e-14) {
		tmp = t_3;
	} else if (x <= -1.3e-82) {
		tmp = 1.0 + (z * (y - (y / x)));
	} else if (x <= -1.35e-131) {
		tmp = t_2;
	} else if (x <= -4.6e-159) {
		tmp = t_1;
	} else if (x <= -4.9e-242) {
		tmp = t_2;
	} else if (x <= 1.25e-109) {
		tmp = y / t;
	} else if (x <= 7.8e-39) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = 1.0d0 - ((y * z) / x)
    t_2 = x * (1.0d0 + ((-1.0d0) / (z * t)))
    t_3 = x / (x + 1.0d0)
    if (x <= (-4.8d-14)) then
        tmp = t_3
    else if (x <= (-1.3d-82)) then
        tmp = 1.0d0 + (z * (y - (y / x)))
    else if (x <= (-1.35d-131)) then
        tmp = t_2
    else if (x <= (-4.6d-159)) then
        tmp = t_1
    else if (x <= (-4.9d-242)) then
        tmp = t_2
    else if (x <= 1.25d-109) then
        tmp = y / t
    else if (x <= 7.8d-39) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - ((y * z) / x);
	double t_2 = x * (1.0 + (-1.0 / (z * t)));
	double t_3 = x / (x + 1.0);
	double tmp;
	if (x <= -4.8e-14) {
		tmp = t_3;
	} else if (x <= -1.3e-82) {
		tmp = 1.0 + (z * (y - (y / x)));
	} else if (x <= -1.35e-131) {
		tmp = t_2;
	} else if (x <= -4.6e-159) {
		tmp = t_1;
	} else if (x <= -4.9e-242) {
		tmp = t_2;
	} else if (x <= 1.25e-109) {
		tmp = y / t;
	} else if (x <= 7.8e-39) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - ((y * z) / x)
	t_2 = x * (1.0 + (-1.0 / (z * t)))
	t_3 = x / (x + 1.0)
	tmp = 0
	if x <= -4.8e-14:
		tmp = t_3
	elif x <= -1.3e-82:
		tmp = 1.0 + (z * (y - (y / x)))
	elif x <= -1.35e-131:
		tmp = t_2
	elif x <= -4.6e-159:
		tmp = t_1
	elif x <= -4.9e-242:
		tmp = t_2
	elif x <= 1.25e-109:
		tmp = y / t
	elif x <= 7.8e-39:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(Float64(y * z) / x))
	t_2 = Float64(x * Float64(1.0 + Float64(-1.0 / Float64(z * t))))
	t_3 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -4.8e-14)
		tmp = t_3;
	elseif (x <= -1.3e-82)
		tmp = Float64(1.0 + Float64(z * Float64(y - Float64(y / x))));
	elseif (x <= -1.35e-131)
		tmp = t_2;
	elseif (x <= -4.6e-159)
		tmp = t_1;
	elseif (x <= -4.9e-242)
		tmp = t_2;
	elseif (x <= 1.25e-109)
		tmp = Float64(y / t);
	elseif (x <= 7.8e-39)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - ((y * z) / x);
	t_2 = x * (1.0 + (-1.0 / (z * t)));
	t_3 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -4.8e-14)
		tmp = t_3;
	elseif (x <= -1.3e-82)
		tmp = 1.0 + (z * (y - (y / x)));
	elseif (x <= -1.35e-131)
		tmp = t_2;
	elseif (x <= -4.6e-159)
		tmp = t_1;
	elseif (x <= -4.9e-242)
		tmp = t_2;
	elseif (x <= 1.25e-109)
		tmp = y / t;
	elseif (x <= 7.8e-39)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(1.0 + N[(-1.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e-14], t$95$3, If[LessEqual[x, -1.3e-82], N[(1.0 + N[(z * N[(y - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.35e-131], t$95$2, If[LessEqual[x, -4.6e-159], t$95$1, If[LessEqual[x, -4.9e-242], t$95$2, If[LessEqual[x, 1.25e-109], N[(y / t), $MachinePrecision], If[LessEqual[x, 7.8e-39], t$95$1, t$95$3]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -4.8e-14 or 7.80000000000000059e-39 < x

   1. Initial program 83.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 83.7%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 83.1%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   6. Step-by-step derivation

      1. +-commutative 83.1%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   7. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if -4.8e-14 < x < -1.3e-82

   1. Initial program 99.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 63.5%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 64.1%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 64.1%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 64.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 64.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 64.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 64.1%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 64.1%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 64.1%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 64.1%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 64.1%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 64.1%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 64.1%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 64.1%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 64.1%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 64.1%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   9. Taylor expanded in x around 0 64.1%

      \[\leadsto 1 + z \cdot \color{blue}{\left(y + -1 \cdot \frac{y}{x}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 64.1%

          \[\leadsto 1 + z \cdot \left(y + \color{blue}{\left(-\frac{y}{x}\right)}\right) \]

       2. unsub-neg 64.1%

          \[\leadsto 1 + z \cdot \color{blue}{\left(y - \frac{y}{x}\right)} \]

   11. Simplified 64.1%

       \[\leadsto 1 + z \cdot \color{blue}{\left(y - \frac{y}{x}\right)} \]

   if -1.3e-82 < x < -1.35000000000000011e-131 or -4.59999999999999957e-159 < x < -4.90000000000000002e-242

   1. Initial program 89.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 89.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 64.1%

      \[\leadsto \frac{\color{blue}{x - \frac{x}{t \cdot z - x}}}{x + 1} \]

   6. Taylor expanded in x around 0 57.3%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{t \cdot z}\right)} \]

   if -1.35000000000000011e-131 < x < -4.59999999999999957e-159 or 1.25000000000000005e-109 < x < 7.80000000000000059e-39

   1. Initial program 96.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 47.0%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 62.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 62.5%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 62.5%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 62.5%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 62.5%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 47.2%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 47.2%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 47.2%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 47.2%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 47.2%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 47.2%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 47.2%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 47.2%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 47.2%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 47.2%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   9. Taylor expanded in x around 0 62.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 62.5%

          \[\leadsto 1 + \color{blue}{\frac{-1 \cdot \left(y \cdot z\right)}{x}} \]

       2. neg-mul-1 62.5%

          \[\leadsto 1 + \frac{\color{blue}{-y \cdot z}}{x} \]

       3. distribute-rgt-neg-in 62.5%

          \[\leadsto 1 + \frac{\color{blue}{y \cdot \left(-z\right)}}{x} \]

   11. Simplified 62.5%

       \[\leadsto 1 + \color{blue}{\frac{y \cdot \left(-z\right)}{x}} \]

   if -4.90000000000000002e-242 < x < 1.25000000000000005e-109

   1. Initial program 85.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 85.9%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 85.9%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 82.1%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 82.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 82.0%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 82.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 82.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   10. Taylor expanded in x around 0 70.1%

       \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-82}:\\ \;\;\;\;1 + z \cdot \left(y - \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-159}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-242}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-39}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{y \cdot z}{x}\\ t_2 := \frac{x}{x + 1} + \frac{y}{t}\\ \mathbf{if}\;x \leq -2.75 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-131}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+58}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (* y z) x))) (t_2 (+ (/ x (+ x 1.0)) (/ y t))))
   (if (<= x -2.75e-18)
     1.0
     (if (<= x -1.35e-131)
       t_2
       (if (<= x -1.2e-140)
         t_1
         (if (<= x 1.05e-61)
           t_2
           (if (<= x 3e-35) t_1 (if (<= x 8.6e+58) t_2 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - ((y * z) / x);
	double t_2 = (x / (x + 1.0)) + (y / t);
	double tmp;
	if (x <= -2.75e-18) {
		tmp = 1.0;
	} else if (x <= -1.35e-131) {
		tmp = t_2;
	} else if (x <= -1.2e-140) {
		tmp = t_1;
	} else if (x <= 1.05e-61) {
		tmp = t_2;
	} else if (x <= 3e-35) {
		tmp = t_1;
	} else if (x <= 8.6e+58) {
		tmp = t_2;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 - ((y * z) / x)
    t_2 = (x / (x + 1.0d0)) + (y / t)
    if (x <= (-2.75d-18)) then
        tmp = 1.0d0
    else if (x <= (-1.35d-131)) then
        tmp = t_2
    else if (x <= (-1.2d-140)) then
        tmp = t_1
    else if (x <= 1.05d-61) then
        tmp = t_2
    else if (x <= 3d-35) then
        tmp = t_1
    else if (x <= 8.6d+58) then
        tmp = t_2
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - ((y * z) / x);
	double t_2 = (x / (x + 1.0)) + (y / t);
	double tmp;
	if (x <= -2.75e-18) {
		tmp = 1.0;
	} else if (x <= -1.35e-131) {
		tmp = t_2;
	} else if (x <= -1.2e-140) {
		tmp = t_1;
	} else if (x <= 1.05e-61) {
		tmp = t_2;
	} else if (x <= 3e-35) {
		tmp = t_1;
	} else if (x <= 8.6e+58) {
		tmp = t_2;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - ((y * z) / x)
	t_2 = (x / (x + 1.0)) + (y / t)
	tmp = 0
	if x <= -2.75e-18:
		tmp = 1.0
	elif x <= -1.35e-131:
		tmp = t_2
	elif x <= -1.2e-140:
		tmp = t_1
	elif x <= 1.05e-61:
		tmp = t_2
	elif x <= 3e-35:
		tmp = t_1
	elif x <= 8.6e+58:
		tmp = t_2
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(Float64(y * z) / x))
	t_2 = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / t))
	tmp = 0.0
	if (x <= -2.75e-18)
		tmp = 1.0;
	elseif (x <= -1.35e-131)
		tmp = t_2;
	elseif (x <= -1.2e-140)
		tmp = t_1;
	elseif (x <= 1.05e-61)
		tmp = t_2;
	elseif (x <= 3e-35)
		tmp = t_1;
	elseif (x <= 8.6e+58)
		tmp = t_2;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - ((y * z) / x);
	t_2 = (x / (x + 1.0)) + (y / t);
	tmp = 0.0;
	if (x <= -2.75e-18)
		tmp = 1.0;
	elseif (x <= -1.35e-131)
		tmp = t_2;
	elseif (x <= -1.2e-140)
		tmp = t_1;
	elseif (x <= 1.05e-61)
		tmp = t_2;
	elseif (x <= 3e-35)
		tmp = t_1;
	elseif (x <= 8.6e+58)
		tmp = t_2;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.75e-18], 1.0, If[LessEqual[x, -1.35e-131], t$95$2, If[LessEqual[x, -1.2e-140], t$95$1, If[LessEqual[x, 1.05e-61], t$95$2, If[LessEqual[x, 3e-35], t$95$1, If[LessEqual[x, 8.6e+58], t$95$2, 1.0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.75e-18 or 8.59999999999999982e58 < x

   1. Initial program 83.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 83.4%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 83.4%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 72.1%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 72.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 72.0%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 72.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 72.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 72.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   10. Taylor expanded in x around inf 86.9%

       \[\leadsto \frac{1}{\color{blue}{1}} \]

   if -2.75e-18 < x < -1.35000000000000011e-131 or -1.19999999999999993e-140 < x < 1.05e-61 or 2.99999999999999989e-35 < x < 8.59999999999999982e58

   1. Initial program 88.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 88.6%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 77.0%

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]

   6. Taylor expanded in t around inf 77.3%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 77.3%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} + \frac{y}{t \cdot \left(1 + x\right)} \]

      2. +-commutative 77.3%

         \[\leadsto \frac{x}{x + 1} + \frac{y}{t \cdot \color{blue}{\left(x + 1\right)}} \]

   8. Simplified 77.3%

      \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}} \]

   9. Taylor expanded in x around 0 75.2%

      \[\leadsto \frac{x}{x + 1} + \frac{y}{\color{blue}{t}} \]

   if -1.35000000000000011e-131 < x < -1.19999999999999993e-140 or 1.05e-61 < x < 2.99999999999999989e-35

   1. Initial program 99.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 70.3%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 85.6%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 85.6%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 85.6%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 85.6%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 85.6%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 69.8%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 69.8%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 69.8%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 69.8%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 69.8%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 69.8%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 69.8%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 69.8%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 69.8%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 69.8%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   9. Taylor expanded in x around 0 85.6%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 85.6%

          \[\leadsto 1 + \color{blue}{\frac{-1 \cdot \left(y \cdot z\right)}{x}} \]

       2. neg-mul-1 85.6%

          \[\leadsto 1 + \frac{\color{blue}{-y \cdot z}}{x} \]

       3. distribute-rgt-neg-in 85.6%

          \[\leadsto 1 + \frac{\color{blue}{y \cdot \left(-z\right)}}{x} \]

   11. Simplified 85.6%

       \[\leadsto 1 + \color{blue}{\frac{y \cdot \left(-z\right)}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-140}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-35}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1}\\ t_2 := 1 - z \cdot \frac{y}{x}\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-172}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 1.03 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-25}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ x 1.0))) (t_2 (- 1.0 (* z (/ y x)))))
   (if (<= x -2.65e-14)
     t_1
     (if (<= x -1.02e-81)
       t_2
       (if (<= x -7.6e-172)
         (* x (/ 1.0 (+ x 1.0)))
         (if (<= x 1.03e-109)
           (/ y t)
           (if (<= x 7.5e-25) t_2 (if (<= x 1.55e-19) (/ y t) t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double t_2 = 1.0 - (z * (y / x));
	double tmp;
	if (x <= -2.65e-14) {
		tmp = t_1;
	} else if (x <= -1.02e-81) {
		tmp = t_2;
	} else if (x <= -7.6e-172) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 1.03e-109) {
		tmp = y / t;
	} else if (x <= 7.5e-25) {
		tmp = t_2;
	} else if (x <= 1.55e-19) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / (x + 1.0d0)
    t_2 = 1.0d0 - (z * (y / x))
    if (x <= (-2.65d-14)) then
        tmp = t_1
    else if (x <= (-1.02d-81)) then
        tmp = t_2
    else if (x <= (-7.6d-172)) then
        tmp = x * (1.0d0 / (x + 1.0d0))
    else if (x <= 1.03d-109) then
        tmp = y / t
    else if (x <= 7.5d-25) then
        tmp = t_2
    else if (x <= 1.55d-19) then
        tmp = y / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double t_2 = 1.0 - (z * (y / x));
	double tmp;
	if (x <= -2.65e-14) {
		tmp = t_1;
	} else if (x <= -1.02e-81) {
		tmp = t_2;
	} else if (x <= -7.6e-172) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= 1.03e-109) {
		tmp = y / t;
	} else if (x <= 7.5e-25) {
		tmp = t_2;
	} else if (x <= 1.55e-19) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (x + 1.0)
	t_2 = 1.0 - (z * (y / x))
	tmp = 0
	if x <= -2.65e-14:
		tmp = t_1
	elif x <= -1.02e-81:
		tmp = t_2
	elif x <= -7.6e-172:
		tmp = x * (1.0 / (x + 1.0))
	elif x <= 1.03e-109:
		tmp = y / t
	elif x <= 7.5e-25:
		tmp = t_2
	elif x <= 1.55e-19:
		tmp = y / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x + 1.0))
	t_2 = Float64(1.0 - Float64(z * Float64(y / x)))
	tmp = 0.0
	if (x <= -2.65e-14)
		tmp = t_1;
	elseif (x <= -1.02e-81)
		tmp = t_2;
	elseif (x <= -7.6e-172)
		tmp = Float64(x * Float64(1.0 / Float64(x + 1.0)));
	elseif (x <= 1.03e-109)
		tmp = Float64(y / t);
	elseif (x <= 7.5e-25)
		tmp = t_2;
	elseif (x <= 1.55e-19)
		tmp = Float64(y / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (x + 1.0);
	t_2 = 1.0 - (z * (y / x));
	tmp = 0.0;
	if (x <= -2.65e-14)
		tmp = t_1;
	elseif (x <= -1.02e-81)
		tmp = t_2;
	elseif (x <= -7.6e-172)
		tmp = x * (1.0 / (x + 1.0));
	elseif (x <= 1.03e-109)
		tmp = y / t;
	elseif (x <= 7.5e-25)
		tmp = t_2;
	elseif (x <= 1.55e-19)
		tmp = y / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.65e-14], t$95$1, If[LessEqual[x, -1.02e-81], t$95$2, If[LessEqual[x, -7.6e-172], N[(x * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.03e-109], N[(y / t), $MachinePrecision], If[LessEqual[x, 7.5e-25], t$95$2, If[LessEqual[x, 1.55e-19], N[(y / t), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.6500000000000001e-14 or 1.5499999999999999e-19 < x

   1. Initial program 83.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 83.4%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 83.4%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 84.3%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   6. Step-by-step derivation

      1. +-commutative 84.3%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   7. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if -2.6500000000000001e-14 < x < -1.01999999999999998e-81 or 1.0300000000000001e-109 < x < 7.49999999999999989e-25

   1. Initial program 99.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 55.9%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 63.8%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 63.8%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 63.8%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 63.8%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 63.8%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 56.7%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 56.7%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 56.7%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 56.7%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 56.7%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 56.7%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 56.7%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 56.7%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 56.7%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 56.7%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   9. Taylor expanded in x around 0 63.8%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}} \]
   10. Step-by-step derivation

       1. mul-1-neg 63.8%

          \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x}\right)} \]

       2. *-commutative 63.8%

          \[\leadsto 1 + \left(-\frac{\color{blue}{z \cdot y}}{x}\right) \]

       3. associate-*r/ 56.7%

          \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x}}\right) \]

       4. distribute-rgt-neg-in 56.7%

          \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x}\right)} \]

       5. distribute-neg-frac2 56.7%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{y}{-x}} \]

   11. Simplified 56.7%

       \[\leadsto 1 + \color{blue}{z \cdot \frac{y}{-x}} \]

   if -1.01999999999999998e-81 < x < -7.59999999999999974e-172

   1. Initial program 87.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 87.4%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 45.2%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   6. Step-by-step derivation

      1. +-commutative 45.2%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   7. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]
   8. Step-by-step derivation

      1. div-inv 45.2%

         \[\leadsto \color{blue}{x \cdot \frac{1}{x + 1}} \]

   9. Applied egg-rr 45.2%

      \[\leadsto \color{blue}{x \cdot \frac{1}{x + 1}} \]

   if -7.59999999999999974e-172 < x < 1.0300000000000001e-109 or 7.49999999999999989e-25 < x < 1.5499999999999999e-19

   1. Initial program 87.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 87.4%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 80.4%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 80.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 80.3%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 80.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 80.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   10. Taylor expanded in x around 0 64.4%

       \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-81}:\\ \;\;\;\;1 - z \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-172}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 1.03 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-25}:\\ \;\;\;\;1 - z \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{z \cdot t - x}}{x + 1}\\ t_2 := \frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+120}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-239}:\\ \;\;\;\;\frac{x + \frac{x - y \cdot z}{x}}{x + 1}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) (- (* z t) x))) (+ x 1.0)))
        (t_2 (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0))))))
   (if (<= z -8.2e+120)
     t_2
     (if (<= z -2e-158)
       t_1
       (if (<= z 5.5e-239)
         (/ (+ x (/ (- x (* y z)) x)) (+ x 1.0))
         (if (<= z 3.3e+128) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((y * z) / ((z * t) - x))) / (x + 1.0);
	double t_2 = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	double tmp;
	if (z <= -8.2e+120) {
		tmp = t_2;
	} else if (z <= -2e-158) {
		tmp = t_1;
	} else if (z <= 5.5e-239) {
		tmp = (x + ((x - (y * z)) / x)) / (x + 1.0);
	} else if (z <= 3.3e+128) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + ((y * z) / ((z * t) - x))) / (x + 1.0d0)
    t_2 = (x / (x + 1.0d0)) + (y / (t * (x + 1.0d0)))
    if (z <= (-8.2d+120)) then
        tmp = t_2
    else if (z <= (-2d-158)) then
        tmp = t_1
    else if (z <= 5.5d-239) then
        tmp = (x + ((x - (y * z)) / x)) / (x + 1.0d0)
    else if (z <= 3.3d+128) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x + ((y * z) / ((z * t) - x))) / (x + 1.0);
	double t_2 = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	double tmp;
	if (z <= -8.2e+120) {
		tmp = t_2;
	} else if (z <= -2e-158) {
		tmp = t_1;
	} else if (z <= 5.5e-239) {
		tmp = (x + ((x - (y * z)) / x)) / (x + 1.0);
	} else if (z <= 3.3e+128) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + ((y * z) / ((z * t) - x))) / (x + 1.0)
	t_2 = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	tmp = 0
	if z <= -8.2e+120:
		tmp = t_2
	elif z <= -2e-158:
		tmp = t_1
	elif z <= 5.5e-239:
		tmp = (x + ((x - (y * z)) / x)) / (x + 1.0)
	elif z <= 3.3e+128:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	t_2 = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * Float64(x + 1.0))))
	tmp = 0.0
	if (z <= -8.2e+120)
		tmp = t_2;
	elseif (z <= -2e-158)
		tmp = t_1;
	elseif (z <= 5.5e-239)
		tmp = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / x)) / Float64(x + 1.0));
	elseif (z <= 3.3e+128)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + ((y * z) / ((z * t) - x))) / (x + 1.0);
	t_2 = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	tmp = 0.0;
	if (z <= -8.2e+120)
		tmp = t_2;
	elseif (z <= -2e-158)
		tmp = t_1;
	elseif (z <= 5.5e-239)
		tmp = (x + ((x - (y * z)) / x)) / (x + 1.0);
	elseif (z <= 3.3e+128)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.2e+120], t$95$2, If[LessEqual[z, -2e-158], t$95$1, If[LessEqual[z, 5.5e-239], N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e+128], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.2e120 or 3.3000000000000001e128 < z

   1. Initial program 67.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 67.9%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 67.9%

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]

   6. Taylor expanded in t around inf 89.8%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 89.8%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} + \frac{y}{t \cdot \left(1 + x\right)} \]

      2. +-commutative 89.8%

         \[\leadsto \frac{x}{x + 1} + \frac{y}{t \cdot \color{blue}{\left(x + 1\right)}} \]

   8. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}} \]

   if -8.2e120 < z < -2.00000000000000013e-158 or 5.49999999999999978e-239 < z < 3.3000000000000001e128

   1. Initial program 96.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 96.5%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 90.6%

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]

   if -2.00000000000000013e-158 < z < 5.49999999999999978e-239

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 88.3%

      \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{-1 \cdot x}}}{x + 1} \]
   6. Step-by-step derivation

      1. neg-mul-1 88.3%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{-x}}}{x + 1} \]

   7. Simplified 88.3%

      \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{-x}}}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{z \cdot t - x}}{x + 1}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-239}:\\ \;\;\;\;\frac{x + \frac{x - y \cdot z}{x}}{x + 1}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+128}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{t}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{t\_1}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{x - y \cdot \frac{z}{x}}{x + 1}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-174} \lor \neg \left(t \leq 5.5 \cdot 10^{-110}\right):\\ \;\;\;\;\frac{t\_1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z \cdot \left(t - y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y t))))
   (if (<= t -2.8e-126)
     (/ 1.0 (/ (+ x 1.0) t_1))
     (if (<= t 5.5e-192)
       (/ (- x (* y (/ z x))) (+ x 1.0))
       (if (or (<= t 4.4e-174) (not (<= t 5.5e-110)))
         (/ t_1 (+ x 1.0))
         (+ 1.0 (/ (* z (- t y)) x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / t);
	double tmp;
	if (t <= -2.8e-126) {
		tmp = 1.0 / ((x + 1.0) / t_1);
	} else if (t <= 5.5e-192) {
		tmp = (x - (y * (z / x))) / (x + 1.0);
	} else if ((t <= 4.4e-174) || !(t <= 5.5e-110)) {
		tmp = t_1 / (x + 1.0);
	} else {
		tmp = 1.0 + ((z * (t - y)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y / t)
    if (t <= (-2.8d-126)) then
        tmp = 1.0d0 / ((x + 1.0d0) / t_1)
    else if (t <= 5.5d-192) then
        tmp = (x - (y * (z / x))) / (x + 1.0d0)
    else if ((t <= 4.4d-174) .or. (.not. (t <= 5.5d-110))) then
        tmp = t_1 / (x + 1.0d0)
    else
        tmp = 1.0d0 + ((z * (t - y)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (y / t);
	double tmp;
	if (t <= -2.8e-126) {
		tmp = 1.0 / ((x + 1.0) / t_1);
	} else if (t <= 5.5e-192) {
		tmp = (x - (y * (z / x))) / (x + 1.0);
	} else if ((t <= 4.4e-174) || !(t <= 5.5e-110)) {
		tmp = t_1 / (x + 1.0);
	} else {
		tmp = 1.0 + ((z * (t - y)) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (y / t)
	tmp = 0
	if t <= -2.8e-126:
		tmp = 1.0 / ((x + 1.0) / t_1)
	elif t <= 5.5e-192:
		tmp = (x - (y * (z / x))) / (x + 1.0)
	elif (t <= 4.4e-174) or not (t <= 5.5e-110):
		tmp = t_1 / (x + 1.0)
	else:
		tmp = 1.0 + ((z * (t - y)) / x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / t))
	tmp = 0.0
	if (t <= -2.8e-126)
		tmp = Float64(1.0 / Float64(Float64(x + 1.0) / t_1));
	elseif (t <= 5.5e-192)
		tmp = Float64(Float64(x - Float64(y * Float64(z / x))) / Float64(x + 1.0));
	elseif ((t <= 4.4e-174) || !(t <= 5.5e-110))
		tmp = Float64(t_1 / Float64(x + 1.0));
	else
		tmp = Float64(1.0 + Float64(Float64(z * Float64(t - y)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (y / t);
	tmp = 0.0;
	if (t <= -2.8e-126)
		tmp = 1.0 / ((x + 1.0) / t_1);
	elseif (t <= 5.5e-192)
		tmp = (x - (y * (z / x))) / (x + 1.0);
	elseif ((t <= 4.4e-174) || ~((t <= 5.5e-110)))
		tmp = t_1 / (x + 1.0);
	else
		tmp = 1.0 + ((z * (t - y)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e-126], N[(1.0 / N[(N[(x + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-192], N[(N[(x - N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 4.4e-174], N[Not[LessEqual[t, 5.5e-110]], $MachinePrecision]], N[(t$95$1 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(z * N[(t - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.79999999999999992e-126

   1. Initial program 81.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 81.4%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 86.9%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 86.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 86.9%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 86.9%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 86.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 86.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   if -2.79999999999999992e-126 < t < 5.49999999999999995e-192

   1. Initial program 90.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 90.9%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 79.1%

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]

   6. Taylor expanded in t around 0 69.9%

      \[\leadsto \color{blue}{\frac{x + -1 \cdot \frac{y \cdot z}{x}}{1 + x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 69.9%

         \[\leadsto \frac{x + \color{blue}{\left(-\frac{y \cdot z}{x}\right)}}{1 + x} \]

      2. unsub-neg 69.9%

         \[\leadsto \frac{\color{blue}{x - \frac{y \cdot z}{x}}}{1 + x} \]

      3. associate-/l* 75.3%

         \[\leadsto \frac{x - \color{blue}{y \cdot \frac{z}{x}}}{1 + x} \]

      4. +-commutative 75.3%

         \[\leadsto \frac{x - y \cdot \frac{z}{x}}{\color{blue}{x + 1}} \]

   8. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{x - y \cdot \frac{z}{x}}{x + 1}} \]

   if 5.49999999999999995e-192 < t < 4.40000000000000043e-174 or 5.4999999999999998e-110 < t

   1. Initial program 86.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 86.8%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 87.5%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

   if 4.40000000000000043e-174 < t < 5.4999999999999998e-110

   1. Initial program 99.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 85.5%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in x around 0 79.7%

      \[\leadsto 1 + \color{blue}{\frac{z \cdot \left(t + -1 \cdot y\right)}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{x - y \cdot \frac{z}{x}}{x + 1}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-174} \lor \neg \left(t \leq 5.5 \cdot 10^{-110}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z \cdot \left(t - y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-85} \lor \neg \left(t \leq 4.6 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + z \cdot \frac{\frac{y}{x}}{-1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.05e-85) (not (<= t 4.6e-102)))
   (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0))))
   (+ 1.0 (* z (/ (/ y x) (- -1.0 x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.05e-85) || !(t <= 4.6e-102)) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else {
		tmp = 1.0 + (z * ((y / x) / (-1.0 - x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.05d-85)) .or. (.not. (t <= 4.6d-102))) then
        tmp = (x / (x + 1.0d0)) + (y / (t * (x + 1.0d0)))
    else
        tmp = 1.0d0 + (z * ((y / x) / ((-1.0d0) - x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.05e-85) || !(t <= 4.6e-102)) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else {
		tmp = 1.0 + (z * ((y / x) / (-1.0 - x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.05e-85) or not (t <= 4.6e-102):
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	else:
		tmp = 1.0 + (z * ((y / x) / (-1.0 - x)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.05e-85) || !(t <= 4.6e-102))
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * Float64(x + 1.0))));
	else
		tmp = Float64(1.0 + Float64(z * Float64(Float64(y / x) / Float64(-1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.05e-85) || ~((t <= 4.6e-102)))
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	else
		tmp = 1.0 + (z * ((y / x) / (-1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.05e-85], N[Not[LessEqual[t, 4.6e-102]], $MachinePrecision]], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(z * N[(N[(y / x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.04999999999999997e-85 or 4.59999999999999973e-102 < t

   1. Initial program 83.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 83.5%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 77.9%

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]

   6. Taylor expanded in t around inf 88.9%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 88.9%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} + \frac{y}{t \cdot \left(1 + x\right)} \]

      2. +-commutative 88.9%

         \[\leadsto \frac{x}{x + 1} + \frac{y}{t \cdot \color{blue}{\left(x + 1\right)}} \]

   8. Simplified 88.9%

      \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}} \]

   if -2.04999999999999997e-85 < t < 4.59999999999999973e-102

   1. Initial program 93.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 93.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 81.0%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 78.3%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 78.3%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 78.3%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 78.3%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 78.3%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 80.9%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 80.9%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 80.9%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 80.9%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 80.9%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 80.9%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 80.9%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 80.9%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 80.9%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 80.9%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-85} \lor \neg \left(t \leq 4.6 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + z \cdot \frac{\frac{y}{x}}{-1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-84} \lor \neg \left(t \leq 6.9 \cdot 10^{-105}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - y \cdot \frac{z}{x}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.7e-84) (not (<= t 6.9e-105)))
   (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0))))
   (/ (- (+ x 1.0) (* y (/ z x))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.7e-84) || !(t <= 6.9e-105)) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else {
		tmp = ((x + 1.0) - (y * (z / x))) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.7d-84)) .or. (.not. (t <= 6.9d-105))) then
        tmp = (x / (x + 1.0d0)) + (y / (t * (x + 1.0d0)))
    else
        tmp = ((x + 1.0d0) - (y * (z / x))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.7e-84) || !(t <= 6.9e-105)) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else {
		tmp = ((x + 1.0) - (y * (z / x))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.7e-84) or not (t <= 6.9e-105):
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	else:
		tmp = ((x + 1.0) - (y * (z / x))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.7e-84) || !(t <= 6.9e-105))
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(Float64(x + 1.0) - Float64(y * Float64(z / x))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.7e-84) || ~((t <= 6.9e-105)))
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	else
		tmp = ((x + 1.0) - (y * (z / x))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.7e-84], N[Not[LessEqual[t, 6.9e-105]], $MachinePrecision]], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.6999999999999999e-84 or 6.90000000000000028e-105 < t

   1. Initial program 83.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 83.5%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 77.9%

      \[\leadsto \frac{x + \color{blue}{\frac{y \cdot z}{t \cdot z - x}}}{x + 1} \]

   6. Taylor expanded in t around inf 88.9%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{y}{t \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 88.9%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} + \frac{y}{t \cdot \left(1 + x\right)} \]

      2. +-commutative 88.9%

         \[\leadsto \frac{x}{x + 1} + \frac{y}{t \cdot \color{blue}{\left(x + 1\right)}} \]

   8. Simplified 88.9%

      \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}} \]

   if -2.6999999999999999e-84 < t < 6.90000000000000028e-105

   1. Initial program 93.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 93.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.3%

      \[\leadsto \color{blue}{\frac{1 + \left(x + -1 \cdot \frac{y \cdot z}{x}\right)}{1 + x}} \]
   6. Step-by-step derivation

      1. associate-+r+ 78.3%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right) + -1 \cdot \frac{y \cdot z}{x}}}{1 + x} \]

      2. mul-1-neg 78.3%

         \[\leadsto \frac{\left(1 + x\right) + \color{blue}{\left(-\frac{y \cdot z}{x}\right)}}{1 + x} \]

      3. unsub-neg 78.3%

         \[\leadsto \frac{\color{blue}{\left(1 + x\right) - \frac{y \cdot z}{x}}}{1 + x} \]

      4. +-commutative 78.3%

         \[\leadsto \frac{\color{blue}{\left(x + 1\right)} - \frac{y \cdot z}{x}}{1 + x} \]

      5. associate-/l* 82.8%

         \[\leadsto \frac{\left(x + 1\right) - \color{blue}{y \cdot \frac{z}{x}}}{1 + x} \]

      6. +-commutative 82.8%

         \[\leadsto \frac{\left(x + 1\right) - y \cdot \frac{z}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - y \cdot \frac{z}{x}}{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-84} \lor \neg \left(t \leq 6.9 \cdot 10^{-105}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - y \cdot \frac{z}{x}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{t}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{t\_1}}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-104}:\\ \;\;\;\;1 + z \cdot \frac{\frac{y}{x}}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y t))))
   (if (<= t -1.3e-77)
     (/ 1.0 (/ (+ x 1.0) t_1))
     (if (<= t 1.08e-104)
       (+ 1.0 (* z (/ (/ y x) (- -1.0 x))))
       (/ t_1 (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / t);
	double tmp;
	if (t <= -1.3e-77) {
		tmp = 1.0 / ((x + 1.0) / t_1);
	} else if (t <= 1.08e-104) {
		tmp = 1.0 + (z * ((y / x) / (-1.0 - x)));
	} else {
		tmp = t_1 / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y / t)
    if (t <= (-1.3d-77)) then
        tmp = 1.0d0 / ((x + 1.0d0) / t_1)
    else if (t <= 1.08d-104) then
        tmp = 1.0d0 + (z * ((y / x) / ((-1.0d0) - x)))
    else
        tmp = t_1 / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (y / t);
	double tmp;
	if (t <= -1.3e-77) {
		tmp = 1.0 / ((x + 1.0) / t_1);
	} else if (t <= 1.08e-104) {
		tmp = 1.0 + (z * ((y / x) / (-1.0 - x)));
	} else {
		tmp = t_1 / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (y / t)
	tmp = 0
	if t <= -1.3e-77:
		tmp = 1.0 / ((x + 1.0) / t_1)
	elif t <= 1.08e-104:
		tmp = 1.0 + (z * ((y / x) / (-1.0 - x)))
	else:
		tmp = t_1 / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / t))
	tmp = 0.0
	if (t <= -1.3e-77)
		tmp = Float64(1.0 / Float64(Float64(x + 1.0) / t_1));
	elseif (t <= 1.08e-104)
		tmp = Float64(1.0 + Float64(z * Float64(Float64(y / x) / Float64(-1.0 - x))));
	else
		tmp = Float64(t_1 / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (y / t);
	tmp = 0.0;
	if (t <= -1.3e-77)
		tmp = 1.0 / ((x + 1.0) / t_1);
	elseif (t <= 1.08e-104)
		tmp = 1.0 + (z * ((y / x) / (-1.0 - x)));
	else
		tmp = t_1 / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.3e-77], N[(1.0 / N[(N[(x + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.08e-104], N[(1.0 + N[(z * N[(N[(y / x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.3000000000000001e-77

   1. Initial program 81.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 81.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 89.2%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 89.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 89.2%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 89.2%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 89.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 89.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   if -1.3000000000000001e-77 < t < 1.07999999999999997e-104

   1. Initial program 93.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 93.0%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 81.0%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 78.3%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 78.3%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 78.3%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 78.3%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 78.3%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 80.9%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 80.9%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 80.9%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 80.9%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 80.9%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 80.9%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 80.9%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 80.9%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 80.9%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 80.9%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   if 1.07999999999999997e-104 < t

   1. Initial program 85.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 85.7%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 87.5%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-104}:\\ \;\;\;\;1 + z \cdot \frac{\frac{y}{x}}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-126} \lor \neg \left(t \leq 5.4 \cdot 10^{-110}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.05e-126) (not (<= t 5.4e-110)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (- 1.0 (/ (* y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.05e-126) || !(t <= 5.4e-110)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - ((y * z) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.05d-126)) .or. (.not. (t <= 5.4d-110))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 - ((y * z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.05e-126) || !(t <= 5.4e-110)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - ((y * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.05e-126) or not (t <= 5.4e-110):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0 - ((y * z) / x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.05e-126) || !(t <= 5.4e-110))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 - Float64(Float64(y * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.05e-126) || ~((t <= 5.4e-110)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0 - ((y * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.05e-126], N[Not[LessEqual[t, 5.4e-110]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.0499999999999999e-126 or 5.3999999999999996e-110 < t

   1. Initial program 83.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 83.1%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 86.7%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

   if -2.0499999999999999e-126 < t < 5.3999999999999996e-110

   1. Initial program 94.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 94.8%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 81.6%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in y around inf 81.1%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 81.1%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y \cdot z}{x \cdot \left(1 + x\right)}\right)} \]

      2. distribute-rgt-in 81.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{1 \cdot x + x \cdot x}}\right) \]

      3. *-lft-identity 81.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{\color{blue}{x} + x \cdot x}\right) \]

      4. unpow2 81.1%

         \[\leadsto 1 + \left(-\frac{y \cdot z}{x + \color{blue}{{x}^{2}}}\right) \]

      5. associate-*l/ 81.5%

         \[\leadsto 1 + \left(-\color{blue}{\frac{y}{x + {x}^{2}} \cdot z}\right) \]

      6. *-commutative 81.5%

         \[\leadsto 1 + \left(-\color{blue}{z \cdot \frac{y}{x + {x}^{2}}}\right) \]

      7. distribute-rgt-neg-in 81.5%

         \[\leadsto 1 + \color{blue}{z \cdot \left(-\frac{y}{x + {x}^{2}}\right)} \]

      8. *-lft-identity 81.5%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{1 \cdot x} + {x}^{2}}\right) \]

      9. unpow2 81.5%

         \[\leadsto 1 + z \cdot \left(-\frac{y}{1 \cdot x + \color{blue}{x \cdot x}}\right) \]

      10. distribute-rgt-in 81.5%

          \[\leadsto 1 + z \cdot \left(-\frac{y}{\color{blue}{x \cdot \left(1 + x\right)}}\right) \]

      11. associate-/r* 81.5%

          \[\leadsto 1 + z \cdot \left(-\color{blue}{\frac{\frac{y}{x}}{1 + x}}\right) \]

      12. distribute-neg-frac 81.5%

          \[\leadsto 1 + z \cdot \color{blue}{\frac{-\frac{y}{x}}{1 + x}} \]

      13. distribute-neg-frac2 81.5%

          \[\leadsto 1 + z \cdot \frac{\color{blue}{\frac{y}{-x}}}{1 + x} \]

   8. Simplified 81.5%

      \[\leadsto 1 + \color{blue}{z \cdot \frac{\frac{y}{-x}}{1 + x}} \]

   9. Taylor expanded in x around 0 72.4%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 72.4%

          \[\leadsto 1 + \color{blue}{\frac{-1 \cdot \left(y \cdot z\right)}{x}} \]

       2. neg-mul-1 72.4%

          \[\leadsto 1 + \frac{\color{blue}{-y \cdot z}}{x} \]

       3. distribute-rgt-neg-in 72.4%

          \[\leadsto 1 + \frac{\color{blue}{y \cdot \left(-z\right)}}{x} \]

   11. Simplified 72.4%

       \[\leadsto 1 + \color{blue}{\frac{y \cdot \left(-z\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-126} \lor \neg \left(t \leq 5.4 \cdot 10^{-110}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-128} \lor \neg \left(t \leq 1.3 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z \cdot \left(t - y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.9e-128) (not (<= t 1.3e-107)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (+ 1.0 (/ (* z (- t y)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.9e-128) || !(t <= 1.3e-107)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 + ((z * (t - y)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-3.9d-128)) .or. (.not. (t <= 1.3d-107))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 + ((z * (t - y)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.9e-128) || !(t <= 1.3e-107)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 + ((z * (t - y)) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.9e-128) or not (t <= 1.3e-107):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0 + ((z * (t - y)) / x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.9e-128) || !(t <= 1.3e-107))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 + Float64(Float64(z * Float64(t - y)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3.9e-128) || ~((t <= 1.3e-107)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0 + ((z * (t - y)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.9e-128], N[Not[LessEqual[t, 1.3e-107]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(z * N[(t - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.89999999999999997e-128 or 1.3e-107 < t

   1. Initial program 83.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 83.1%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 86.7%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

   if -3.89999999999999997e-128 < t < 1.3e-107

   1. Initial program 94.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 94.8%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 81.6%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 \cdot \frac{y}{x \cdot \left(1 + x\right)} + \frac{t}{x \cdot \left(1 + x\right)}\right)} \]

   6. Taylor expanded in x around 0 72.5%

      \[\leadsto 1 + \color{blue}{\frac{z \cdot \left(t + -1 \cdot y\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-128} \lor \neg \left(t \leq 1.3 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z \cdot \left(t - y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-171} \lor \neg \left(x \leq 8 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5e-171) (not (<= x 8e-59))) (/ x (+ x 1.0)) (/ y t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e-171) || !(x <= 8e-59)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = y / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-5d-171)) .or. (.not. (x <= 8d-59))) then
        tmp = x / (x + 1.0d0)
    else
        tmp = y / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e-171) || !(x <= 8e-59)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = y / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5e-171) or not (x <= 8e-59):
		tmp = x / (x + 1.0)
	else:
		tmp = y / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5e-171) || !(x <= 8e-59))
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(y / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5e-171) || ~((x <= 8e-59)))
		tmp = x / (x + 1.0);
	else
		tmp = y / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5e-171], N[Not[LessEqual[x, 8e-59]], $MachinePrecision]], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999992e-171 or 8.0000000000000002e-59 < x

   1. Initial program 85.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 85.7%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 72.2%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
   6. Step-by-step derivation

      1. +-commutative 72.2%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   7. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if -4.99999999999999992e-171 < x < 8.0000000000000002e-59

   1. Initial program 88.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 88.6%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 78.8%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 78.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 78.7%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 78.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 78.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 78.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   10. Taylor expanded in x around 0 60.7%

       \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-171} \lor \neg \left(x \leq 8 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -7e-26) 1.0 (if (<= x 3.8e-54) (/ y t) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7e-26) {
		tmp = 1.0;
	} else if (x <= 3.8e-54) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-7d-26)) then
        tmp = 1.0d0
    else if (x <= 3.8d-54) then
        tmp = y / t
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7e-26) {
		tmp = 1.0;
	} else if (x <= 3.8e-54) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -7e-26:
		tmp = 1.0
	elif x <= 3.8e-54:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7e-26)
		tmp = 1.0;
	elseif (x <= 3.8e-54)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -7e-26)
		tmp = 1.0;
	elseif (x <= 3.8e-54)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -7e-26], 1.0, If[LessEqual[x, 3.8e-54], N[(y / t), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999997e-26 or 3.8000000000000002e-54 < x

   1. Initial program 84.5%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 84.5%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 69.6%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 69.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 69.6%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 69.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 69.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   10. Taylor expanded in x around inf 78.3%

       \[\leadsto \frac{1}{\color{blue}{1}} \]

   if -6.9999999999999997e-26 < x < 3.8000000000000002e-54

   1. Initial program 89.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 89.3%

         \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 73.6%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
   6. Step-by-step derivation

      1. clear-num 73.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. inv-pow 73.6%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

   7. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 73.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

      2. +-commutative 73.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

   9. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

   10. Taylor expanded in x around 0 50.4%

       \[\leadsto \color{blue}{\frac{y}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 3.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ -1.0 x))

C

double code(double x, double y, double z, double t) {
	return -1.0 / x;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-1.0d0) / x
end function

Java

public static double code(double x, double y, double z, double t) {
	return -1.0 / x;
}

Python

def code(x, y, z, t):
	return -1.0 / x

Julia

function code(x, y, z, t)
	return Float64(-1.0 / x)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -1.0 / x;
end

Wolfram

code[x_, y_, z_, t_] := N[(-1.0 / x), $MachinePrecision]

Derivation

1. Initial program 86.6%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
2. Step-by-step derivation

   1. *-commutative 86.6%

      \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

3. Simplified 86.6%

   \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 56.4%

   \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
6. Step-by-step derivation

   1. +-commutative 56.4%

      \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

7. Simplified 56.4%

   \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

8. Taylor expanded in x around inf 43.8%

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]

9. Taylor expanded in x around 0 3.2%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]

10. Final simplification 3.2%

    \[\leadsto \frac{-1}{x} \]
11. Add Preprocessing

Alternative 19: 53.2% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 1.0)

C

double code(double x, double y, double z, double t) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0;
}

Python

def code(x, y, z, t):
	return 1.0

Julia

function code(x, y, z, t)
	return 1.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 86.6%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
2. Step-by-step derivation

   1. *-commutative 86.6%

      \[\leadsto \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

3. Simplified 86.6%

   \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 71.4%

   \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
6. Step-by-step derivation

   1. clear-num 71.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

   2. inv-pow 71.3%

      \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]

7. Applied egg-rr 71.3%

   \[\leadsto \color{blue}{{\left(\frac{x + 1}{x + \frac{y}{t}}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 71.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x + \frac{y}{t}}}} \]

   2. +-commutative 71.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 + x}}{x + \frac{y}{t}}} \]

9. Simplified 71.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 + x}{x + \frac{y}{t}}}} \]

10. Taylor expanded in x around inf 50.0%

    \[\leadsto \frac{1}{\color{blue}{1}} \]

11. Final simplification 50.0%

    \[\leadsto 1 \]
12. Add Preprocessing

Developer target: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0)))

C

double code(double x, double y, double z, double t) {
	return (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0);
}

Python

def code(x, y, z, t):
	return (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(y / Float64(t - Float64(x / z))) - Float64(x / Float64(Float64(t * z) - x)))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(y / N[(t - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024039 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))